A327386 -id:A327386 - cp's OEIS frontend

A308680 Number T(n,k) of colored integer partitions of n such that all colors from a k-set are used and parts differ by size or by color; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 5, 3, 1, 0, 3, 8, 9, 4, 1, 0, 4, 14, 19, 14, 5, 1, 0, 5, 22, 39, 36, 20, 6, 1, 0, 6, 34, 72, 85, 60, 27, 7, 1, 0, 8, 50, 128, 180, 160, 92, 35, 8, 1, 0, 10, 73, 216, 360, 381, 273, 133, 44, 9, 1, 0, 12, 104, 354, 680, 845, 720, 434, 184, 54, 10, 1

Offset: 0

Alois P. Heinz, Aug 29 2019

For fixed k > 0, T(n,k) ~ exp(Pi*sqrt(k*n/3)) * k^(1/4) / (3^(1/4) * 2^((k+3)/2) * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019

T is the convolution triangle of A000009 (see A357368). - Peter Luschny, Oct 19 2022

			T(4,1) = 2: 3a1a, 4a.
T(4,2) = 5: 2a1a1b, 2b1a1b, 2a2b, 3a1b, 3b1a.
T(4,3) = 3: 2a1b1c, 2b1a1c, 2c1a1b.
T(4,4) = 1: 1a1b1c1d.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  2,   1;
  0,  2,  5,   3,   1;
  0,  3,  8,   9,   4,   1;
  0,  4, 14,  19,  14,   5,   1;
  0,  5, 22,  39,  36,  20,   6,   1;
  0,  6, 34,  72,  85,  60,  27,   7,  1;
  0,  8, 50, 128, 180, 160,  92,  35,  8, 1;
  0, 10, 73, 216, 360, 381, 273, 133, 44, 9, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Partition (number theory)

Columns k=0-10 give: A000007, A000009 (for n>0), A327380, A327381, A327382, A327383, A327384, A327385, A327386, A327387, A327388.
Main diagonal and lower diagonals give: A000012, A001477, A000096.
Row sums give A304969.
T(2n,n) gives A324595.
Cf. A060642, A286335, A325915.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
T:= proc(n, k) option remember;
      `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, b(n)),
          (q-> add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Jan 31 2021
# Uses function PMatrix from A357368.
PMatrix(10, A000009); # Peter Luschny, Oct 19 2022

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t,    b[t, Min[t, i - 1], k]*Binomial[k, j]][n - i*j], {j, 0, Min[k, n/i]}]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, from Maple *)

T(n,k) = Sum_{i=0..k} (-1)^i * binomial(k,i) * A286335(n,k-i).
Sum_{k=1..n} k * T(n,k) = A325915(n).
G.f. of column k: (-1 + Product_{j>=1} (1 + x^j))^k. - Alois P. Heinz, Jan 29 2021

A341227 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^8.

Original entry on oeis.org

1, 16, 136, 824, 4004, 16608, 61076, 204200, 631714, 1831752, 5027312, 13159104, 33049090, 80030808, 187613348, 427201176, 947520103, 2051989360, 4347996772, 9030416704, 18412343832, 36905322248, 72807201940, 141525042736, 271321432489, 513454659312

Offset: 8

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..10000

Column k=8 of A060642.

Cf. A000041, A023007, A048574, A327386, A341221, A341222, A341223, A341225, A341226, A341228.

b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 8):
seq(a(n), n=8..33);  # Alois P. Heinz, Feb 07 2021

nmax = 33; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^8, {x, 0, nmax}], x] // Drop[#, 8] &

A341391 Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^8.

Original entry on oeis.org

1, 16, 152, 1072, 6204, 31024, 138544, 564824, 2135902, 7580944, 25485560, 81734696, 251514840, 746123304, 2142114356, 5971477112, 16208165181, 42936937488, 111240873128, 282363615336, 703303327288, 1721329848680, 4144792701532, 9829483710112

Offset: 8

Ilya Gutkovskiy, Feb 10 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..10000

Cf. A026007, A321953, A327386, A341384, A341385, A341386, A341387, A341388, A341390.

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(d^2/
     `if`(d::odd, 1, 2), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 8):
seq(a(n), n=8..31);  # Alois P. Heinz, Feb 10 2021

nmax = 31; CoefficientList[Series[(-1 + Product[(1 + x^k)^k, {k, 1, nmax}])^8, {x, 0, nmax}], x] // Drop[#, 8] &

A341247 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^8.

Original entry on oeis.org

1, 0, 8, 8, 36, 64, 148, 296, 562, 1080, 1920, 3440, 5890, 9992, 16532, 26920, 43175, 68144, 106260, 163472, 248824, 374504, 558212, 824208, 1206409, 1751360, 2522692, 3607456, 5122848, 7227392, 10132948, 14123000, 19573393, 26981768, 37003700, 50499952, 68595956

Offset: 8

Ilya Gutkovskiy, Feb 07 2021

nonn

Cf. A000700, A001486, A007259, A101127, A327386, A338463, A341227, A341241, A341243, A341244, A341245, A341246, A341251.

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 8):
seq(a(n), n=8..44);  # Alois P. Heinz, Feb 07 2021

nmax = 44; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^8, {x, 0, nmax}], x] // Drop[#, 8] &

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^8.

A341369 Expansion of (1 / theta_4(x) - 1)^8 / 256.

Original entry on oeis.org

1, 16, 144, 952, 5136, 23904, 99292, 376512, 1324376, 4372632, 13673888, 40787848, 116713350, 321861312, 858693192, 2223428224, 5602833292, 13772292360, 33089930724, 77846837848, 179602530648, 406914172336, 906438716196, 1987418937952, 4293164981849, 9144987747024

Offset: 8

Ilya Gutkovskiy, Feb 10 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..10000

Cf. A002448, A004409, A014968, A015128, A319553, A327386, A338223, A340915, A341227, A341364, A341365, A341366, A341367, A341368, A341370.

g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
      g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 8):
seq(a(n), n=8..33);  # Alois P. Heinz, Feb 10 2021

nmax = 33; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^8/256, {x, 0, nmax}], x] // Drop[#, 8] &
nmax = 33; CoefficientList[Series[(1/256) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^8, {x, 0, nmax}], x] // Drop[#, 8] &

G.f.: (1/256) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^8.

cp's OEIS Frontend

A308680 Number T(n,k) of colored integer partitions of n such that all colors from a k-set are used and parts differ by size or by color; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A341227 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^8.

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A341391 Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^8.

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A341247 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^8.

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A341369 Expansion of (1 / theta_4(x) - 1)^8 / 256.

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